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Lhspairs

Lhspairs-fix

(lhspairs-fix x) is an fty alist fixing function that follows the fix-keys strategy.

Signature
(lhspairs-fix x) → fty::newx
Arguments: x — Guard (lhspairs-p x).
Returns: fty::newx — Type (lhspairs-p fty::newx).

Note that in the execution this is just an inline identity function.

Definitions and Theorems

Function: lhspairs-fix$inline

(defun lhspairs-fix$inline (x)
  (declare (xargs :guard (lhspairs-p x)))
  (let ((__function__ 'lhspairs-fix))
    (declare (ignorable __function__))
    (mbe :logic
         (if (atom x)
             x
           (if (consp (car x))
               (cons (cons (lhs-fix (caar x))
                           (lhs-fix (cdar x)))
                     (lhspairs-fix (cdr x)))
             (lhspairs-fix (cdr x))))
         :exec x)))

Theorem: lhspairs-p-of-lhspairs-fix

(defthm lhspairs-p-of-lhspairs-fix
  (b* ((fty::newx (lhspairs-fix$inline x)))
    (lhspairs-p fty::newx))
  :rule-classes :rewrite)

Theorem: lhspairs-fix-when-lhspairs-p

(defthm lhspairs-fix-when-lhspairs-p
  (implies (lhspairs-p x)
           (equal (lhspairs-fix x) x)))

Function: lhspairs-equiv$inline

(defun lhspairs-equiv$inline (x y)
  (declare (xargs :guard (and (lhspairs-p x) (lhspairs-p y))))
  (equal (lhspairs-fix x)
         (lhspairs-fix y)))

Theorem: lhspairs-equiv-is-an-equivalence

(defthm lhspairs-equiv-is-an-equivalence
  (and (booleanp (lhspairs-equiv x y))
       (lhspairs-equiv x x)
       (implies (lhspairs-equiv x y)
                (lhspairs-equiv y x))
       (implies (and (lhspairs-equiv x y)
                     (lhspairs-equiv y z))
                (lhspairs-equiv x z)))
  :rule-classes (:equivalence))

Theorem: lhspairs-equiv-implies-equal-lhspairs-fix-1

(defthm lhspairs-equiv-implies-equal-lhspairs-fix-1
  (implies (lhspairs-equiv x x-equiv)
           (equal (lhspairs-fix x)
                  (lhspairs-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: lhspairs-fix-under-lhspairs-equiv

(defthm lhspairs-fix-under-lhspairs-equiv
  (lhspairs-equiv (lhspairs-fix x) x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-lhspairs-fix-1-forward-to-lhspairs-equiv

(defthm equal-of-lhspairs-fix-1-forward-to-lhspairs-equiv
  (implies (equal (lhspairs-fix x) y)
           (lhspairs-equiv x y))
  :rule-classes :forward-chaining)

Theorem: equal-of-lhspairs-fix-2-forward-to-lhspairs-equiv

(defthm equal-of-lhspairs-fix-2-forward-to-lhspairs-equiv
  (implies (equal x (lhspairs-fix y))
           (lhspairs-equiv x y))
  :rule-classes :forward-chaining)

Theorem: lhspairs-equiv-of-lhspairs-fix-1-forward

(defthm lhspairs-equiv-of-lhspairs-fix-1-forward
  (implies (lhspairs-equiv (lhspairs-fix x) y)
           (lhspairs-equiv x y))
  :rule-classes :forward-chaining)

Theorem: lhspairs-equiv-of-lhspairs-fix-2-forward

(defthm lhspairs-equiv-of-lhspairs-fix-2-forward
  (implies (lhspairs-equiv x (lhspairs-fix y))
           (lhspairs-equiv x y))
  :rule-classes :forward-chaining)

Theorem: cons-of-lhs-fix-k-under-lhspairs-equiv

(defthm cons-of-lhs-fix-k-under-lhspairs-equiv
  (lhspairs-equiv (cons (cons (lhs-fix acl2::k) acl2::v)
                        x)
                  (cons (cons acl2::k acl2::v) x)))

Theorem: cons-lhs-equiv-congruence-on-k-under-lhspairs-equiv

(defthm cons-lhs-equiv-congruence-on-k-under-lhspairs-equiv
  (implies (lhs-equiv acl2::k k-equiv)
           (lhspairs-equiv (cons (cons acl2::k acl2::v) x)
                           (cons (cons k-equiv acl2::v) x)))
  :rule-classes :congruence)

Theorem: cons-of-lhs-fix-v-under-lhspairs-equiv

(defthm cons-of-lhs-fix-v-under-lhspairs-equiv
  (lhspairs-equiv (cons (cons acl2::k (lhs-fix acl2::v))
                        x)
                  (cons (cons acl2::k acl2::v) x)))

Theorem: cons-lhs-equiv-congruence-on-v-under-lhspairs-equiv

(defthm cons-lhs-equiv-congruence-on-v-under-lhspairs-equiv
  (implies (lhs-equiv acl2::v v-equiv)
           (lhspairs-equiv (cons (cons acl2::k acl2::v) x)
                           (cons (cons acl2::k v-equiv) x)))
  :rule-classes :congruence)

Theorem: cons-of-lhspairs-fix-y-under-lhspairs-equiv

(defthm cons-of-lhspairs-fix-y-under-lhspairs-equiv
  (lhspairs-equiv (cons x (lhspairs-fix y))
                  (cons x y)))

Theorem: cons-lhspairs-equiv-congruence-on-y-under-lhspairs-equiv

(defthm cons-lhspairs-equiv-congruence-on-y-under-lhspairs-equiv
  (implies (lhspairs-equiv y y-equiv)
           (lhspairs-equiv (cons x y)
                           (cons x y-equiv)))
  :rule-classes :congruence)

Theorem: lhspairs-fix-of-acons

(defthm lhspairs-fix-of-acons
  (equal (lhspairs-fix (cons (cons acl2::a acl2::b) x))
         (cons (cons (lhs-fix acl2::a)
                     (lhs-fix acl2::b))
               (lhspairs-fix x))))

Theorem: lhspairs-fix-of-append

(defthm lhspairs-fix-of-append
  (equal (lhspairs-fix (append std::a std::b))
         (append (lhspairs-fix std::a)
                 (lhspairs-fix std::b))))

Theorem: consp-car-of-lhspairs-fix

(defthm consp-car-of-lhspairs-fix
  (equal (consp (car (lhspairs-fix x)))
         (consp (lhspairs-fix x))))